IN20 · Thickness changes reflected polarization

Spectroscopic Ellipsometer

Can reflected polarization tell film thickness and refractive index apart? Several wavelengths and angles often help separate thickness from refractive index because each changes the reflected polarization differently. A single color can repeat the same response at several thicknesses. Matching the substrate index or strongly absorbing the returning light can leave thickness poorly determined even with a good fit.

Rotate two optical arms, keep intensity scans and reference calibration, then compare competing film fits across wavelengths and angles.

One fitted thickness $119.6\,\mathrm{nm}$

Candidate index $1.500$

Candidate extinction $0.014$

Normalized ratio RMS $5.72\times10^{-4}$

Compare film candidates

Film thickness

d/\mathrm{nm}

120

Film refractive index

n

1.5

Incidence angle

\theta/{}^\circ

65

Source polarizer

P/{}^\circ

45

Central vacuum wavelength

\lambda/\mathrm{nm}

632.8

Drag the film or the rear goniometer; receiving mounts change their physical zero offsets. Recalibrate after changing the receiving optics or wavelengths.

The calibrated default scan is ready.

Acquisition, fit and model

Observation coverageFit parameter familyGenerated extinction

kappa

0.015

Analyzer zero offset

A_0/{}^\circ

2.3

Compensator zero offset

C_0/{}^\circ

-3.1

Retardance at 550 nm

\delta_{550}/{}^\circ

92

Detector gain

g

1.04

Compensator in receiving beamSeed

Six externally characterized unit-Stokes states calibrate every receiver row from raw ADC counts. A different elliptical state checks it. The reference source and sample source share stable flux; no absolute-power accuracy is claimed.

The film has constant index and extinction over this finite band, air incidence and a known lossless substrate of index 1.7. The fit compares complex ratios and preserves competing minima. The cost neighborhood is numerical, not a confidence interval.

Film thickness is enlarged 25,000 times; field glyphs use one common amplitude scale and schematic speed. Replay slows the stored 50 ms exposure steps. Roughness, anisotropy, depolarization, finite bandwidth and drift are omitted.

Model assumptions

A coherent, flat isotropic single layer lies between air and a known lossless substrate of index 1.7. Film index and extinction are constant over the finite measured band; this is a teaching material, not a material database or a roughness model. Harmonic fields use exp of spatial phase minus time phase and right-handed local p/s/propagation bases. Ellipsometric ratio is full-vector p reflection over s reflection; the existing tangential-field p coefficient requires a sign adapter. Twenty-four compensator/analyzer states and six externally characterized unit-Stokes reference states calibrate receiver rows from raw ADC counts; a separate elliptical reference checks them. Stable reference/sample flux, perfect source-polarizer angle and ideal exposure timing are assumed. Sample phase is recovered only with adequate circular sensitivity and reflection. A bounded multi-start fit uses saved complex ratios; competing minima and weak thickness sensitivity remain visible. Candidate neighborhoods are numerical comparisons, not confidence intervals. Film thickness is enlarged 25,000 times; field glyphs and scan replay are slowed. Anisotropy, depolarization, roughness, finite bandwidth and source drift are omitted.

Intensity scans determine the polarization change

Raw central intensity scan

$A/{}^\circ\quad;\quad N_{\mathrm{ADC}}/\mathrm{counts}$

Turquoise: compensator at zero degrees; violet: forty-five degrees. The marker and physical display replay one stored exposure.

Choose incidence angle

Amplitude angle $\Psi$

$\lambda_{\mathrm{vac}}/\mathrm{nm}\quad;\quad\Psi/{}^\circ$

Phase difference $\Delta$

$\lambda_{\mathrm{vac}}/\mathrm{nm}\quad;\quad\Delta/{}^\circ$

Phase wraps leave a gap.

Filled marks come from raw reduction; dashes or open marks show one candidate, with each selected-angle panel scaled independently.

Competing films, reference checks and saved scans

Thickness profile at the candidate optical constants

$d/\mathrm{nm}\quad;\quad\log_{10}L$

L=\frac1N\sum_j\frac{|\rho_{\mathrm{fit},j}-\widehat\rho_j|^2}{1+|\widehat\rho_j|^2}

Distinct numerical candidates
$d/\mathrm{nm}$	$n$	$\kappa$	Cost

The profile holds candidate optical constants fixed; it is not a confidence interval or a full marginalization over correlated parameters. Several starting points reduce local-minimum risk but do not prove global uniqueness.

Frozen raw scans
Record	Conditions	Calibration	Fit

Exports retain all reference/dark/check and sample ADC counts, commanded angles, relative exposure times, seeds, conventions, model family and fitted candidates. Preview changes leave kept raw and processing snapshots unchanged.

Start with the essentials

Focus question: When do several colors and angles distinguish thickness from refractive index?
One-sentence intuition: A measured polarization change constrains an optical model; several wavelengths can separate film candidates that coincide at one color.

Core mathematical model

The complex ellipsometric ratio

\rho=\frac{r_p}{r_s}=\tan\Psi\,e^{\mathrm i\Delta}

The reflection amplitudes use right-handed local p, s and propagation axes. Psi specifies their amplitude ratio and Delta their phase difference. A detector observes intensities at known compensator and analyzer angles.

One transparent-film round trip

\beta=\frac{2\pi d}{\lambda_{\mathrm{vac}}}\sqrt{N^2-\sin^2\theta},\qquad N=n+\mathrm i\kappa

The phase convention is spatial phase minus time phase. A round trip contributes twice beta. For a transparent film, adding a whole round-trip cycle repeats the single-color reflection. Absorption attenuates the returning field.

Common difficulties

A small residual can have several solutions

Typical misconceptionThe lowest displayed cost proves a unique thickness.

Better mental modelThe film model may contain several similar minima or a broad weak-sensitivity region. The numerical comparison tolerance is not a confidence interval.

A linear analyzer can lose part of the phase

Typical misconceptionRotating a single polarizer always gives the full polarization ellipse.

Better mental modelThe compensator converts circular sensitivity into a measurable intensity change. Parking it can make the reference matrix rank deficient.

Run the experiment

01
Watch one exposure
Pause the replay and compare the physical analyzer display with the marker on the raw scan.
What to observe: Both refer to the same stored detector count and commanded analyzer angle.
02
Compare one color with several
Choose the repeated-thickness preset, then restore several wavelengths and angles.
What to observe: Several thicknesses can fit a single color. Additional conditions usually separate those branches.
03
Change the receiving optics
Drag the compensator or analyzer mount to change its zero offset, then measure new reference states.
What to observe: A hardware change invalidates the old calibration. Six known input states calibrate the receiver; a different ellipse checks it.
04
Remove the lower reflection
Compare the index-matched and strongly absorbing presets.
What to observe: A film can become insensitive to thickness because the lower interface vanishes or its returning light becomes too weak.

Spectroscopic Ellipsometer

Intensity scans determine the polarization change

Raw central intensity scan

Choose incidence angle

Amplitude angle $\Psi$

Phase difference $\Delta$

Thickness profile at the candidate optical constants

Read the phase before choosing a film

Core mathematical model

The complex ellipsometric ratio

One transparent-film round trip

Common difficulties

A small residual can have several solutions

A linear analyzer can lose part of the phase

Run the experiment

Watch one exposure

Compare one color with several

Change the receiving optics

Remove the lower reflection

Related experiments and articles

How physics instruments measure

Intensity scans determine the polarization change

Raw central intensity scan

Choose incidence angle

Amplitude angle Ψ\PsiΨ

Phase difference Δ\DeltaΔ

Thickness profile at the candidate optical constants

Core mathematical model

The complex ellipsometric ratio

One transparent-film round trip

Common difficulties

A small residual can have several solutions

A linear analyzer can lose part of the phase

Run the experiment

Watch one exposure

Compare one color with several

Change the receiving optics

Remove the lower reflection

Related experiments and articles

How physics instruments measure

Amplitude angle $\Psi$

Phase difference $\Delta$